Risk Distribution

Definition

A Risk Distribution is the core computational tool (building block) of Quantitative Risk Management. Mathematically a risk distribution is the Probability Distribution of a Random Variable.

Risk Distribution Categories

One can classify risk distributions along a number of interesting characteristics and these tend to be very important in determining fitness of use for particular risk applications

• Parametric or non-parametric
• Obtained as an Analytic Models of via Monte-Carlo Simulation
• Discrete, continuous or mixed support. The nature of the realized risk usually dictates which type is meaningful to use
• Finite, infinite one sided or infinite two sided support. Similarly as above
• Finite of infinite moments / moment generating function. Related to the sub-exponential, heavy-tails and fat-tails properties. NB: It may not always be possible to assert whether an actual phenomenon has infinite moments
• Symmetric or asymmetric.

Usage

• The most direct application of risk distributions in risk management is the explicit modelling of risk as realisations from a distribution
• The concept forms the basis for the constructions of a variety of risk metrics.
• It is rare that a risk phenomenon is described exclusively by a single parametric distribution. Hence many usage examples are cases where a distribution is used as part of an overall modelling framework (which is multi-dimensional and/or involves time-dependent processes)
• Some distributions (e.g, the uniform and the normal) are used extensively as building blocks to create more realistic distributions
• Some distributions appear primarily in the context of statistical testing that is used in risk management applications
• While in the majority of applications distributions are used in conjuction with historical data (models of risk realisations), in financial applications involving the pricing of risk they may also be used as the implied distributions (of potential future risk events)

Usage by Risk Type

The quantitative risk management of all major risk types involves risk distributions of various types

• Credit Risk Loss Distribution Function
• Market Risk
• Operational Risk
• Business Model Risk

Risk Distribution Catalog

The catalog aims to be a comprehensive list of one dimensional (univariate), parametric risk distributions as they are used in risk management applications. More general (non-parametric) approaches are out of scope.

The underlying probability distribution functions are generally well documented in Wikipedia therefore the table below links to individual entries for further information on the functional form and other mathematical properties of each distribution. The main purpose of the catalog is establish an overview of aspects that are interesting in risk context

Some distributions may coincide for certain parameters or be special cases

The references are not meant to be exhaustive nor setting academic priority, they are merely easily available examples of usage.

Name	Discreteness	Boundedness	Finite Moments	Symmetry	Comments	Market Risk	Credit Risk	Operational / Insurance Risk
Beta	Continuous	Finite	All	No (unless a=b)	Widely used as a model for a fractional (percent) outcome		Loss-given-default model
Bernoulli	Discrete	Finite	All	No (unless p=0.5)	Widely used as the archetype of coin toss experiment (n=2)	Tree based pricing of options	Default / No-default model
Binomial	Discrete	Finite	All	No (unless p=0.5)	Counts number of events in a sequence of n independent experiments	Value-at-risk exceptions	Simple model of portfolio default count
Cauchy	Continuous	Infinite	None	No (for specific parameters)	Example of distribution with no defined expectation (therefore difficult to use in risk management)
Chi-squared	Continuous	Infinite	All	No	Sum of squared Normal variables		Used in credit scorecard validation
Dirac Delta	Continuous	Infinite	All	Yes	A degenate distribution serving as a link between the continuous and discrete categories
Exponential	Continuous	Infinite	All	No	Archetype of event arrival time (e.g for a Poisson process)	As component in market jump processes	As credit event arrival time model	As operational risk event arrival time
F Distribution	Continuous	Infinite	3	No	Arrises in statistical testing of variance (F-Test)		In Model Validation
Gamma	Continuous	Infinite	All	No	Generalization of exponential, chi-squared etc distributions			As operational risk event severity model
Gumbel	Continuous	Infinite	All	No	The distribution of the extreme realizations when sampling from various distributions (Extreme Value Theory)
Inverse Gaussian	Continuous	Infinite	All	No		First passage time of Brownian Motion		As operational risk event severity model
Log-Normal	Continuous	Infinite	All	No	The product of many independent random variables, each of which is positive	Archetypal model for asset prices	Used in threshold (distance-to-default) credit models	As operational risk event severity model
Logistic	Continuous	Infinite	All	No	Its cumulative distribution function is the logistic function		Widely used (implicitly) in logistic regression models
Normal (Gaussian)	Continuous	Infinite	All	Yes	The most commonly used distribution function due to its expectional tractability, limit theorems and natural emergence in physical phenomena	The basis of large fraction of stochastic processes	Implicit in many statistical analyses
Poisson	Discrete	Infinite	All	No	The probability of a given number of events occurring in a fixed interval of time	As component in market jump processes	Credit event count	Operational event count
Student's t	Continuous	Infinite	All	Yes	Arrises in statistical testing (Student's t Test)		In Model Validation
Uniform	Continuous	Finite	All	Yes	Used as building block for generating samples from other distrubutions
Vasicek Distribution	Continuous	Finite	All	No	Arrises as a limit distribution of the sum of conditionally independent Bernulli variables		Used as a simple model of portfolio loss (also in Basel II as the ASRF model)
Weibull	Continuous	Infinite	All	No	A generalization of the exponential		As parametric model of survival curves	As operational risk event severity

Issues and Challenges

• The availability of a very large number of parametric risk distributions means that modelling univariate risks becomes often a mechanical exercise focusing of measures of fit without regard as to whether the distribution has some intrinsic affinity with the process being modelled

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